Optical manipulation of Rashba-split 2-dimensional electron gas

In spintronics, the two main approaches to actively control the electrons’ spin involve static magnetic or electric fields. An alternative avenue relies on the use of optical fields to generate spin currents, which can bolster spin-device performance, allowing for faster and more efficient logic. To date, research has mainly focused on the optical injection of spin currents through the photogalvanic effect, and little is known about the direct optical control of the intrinsic spin-splitting. To explore the optical manipulation of a material’s spin properties, we consider the Rashba effect. Using time- and angle-resolved photoemission spectroscopy (TR-ARPES), we demonstrate that an optical excitation can tune the Rashba-induced spin splitting of a two-dimensional electron gas at the surface of Bi2Se3. We establish that light-induced photovoltage and charge carrier redistribution - which in concert modulate the Rashba spin-orbit coupling strength on a sub-picosecond timescale - can offer an unprecedented platform for achieving optically-driven spin logic devices.

here we highlight the phenomenological implications and the corresponding data treatment process.
The sign of E P V depends on the equilibrium band bending of the material. For a downward band bending -as it is the case in our experiment -E P V in the vacuum region points toward the surface, resulting in an outward accelerating force F P V acting on the negative charges (electrons) outside the material. A schematic of this effect is presented in Fig. S1a, showcasing the effect of the photovoltage on the time-resolved photoemission process at positive and negative time delays.
In the left panel of Fig. S1a (positive delays), the pump pulse comes first, establishing the photovoltaic field in the area of the sample illuminated. Subsequently, the probe pulse emits an electron from the pumped area, and its kinetic energy is increased by the PV field. The kinetic energy of the photo-electron at positive delays is a reflection of its binding energy inside the material as well as the magnitude of the PV. For a simple surface state, this manifests as a rigid shift of the photoemission spectrum with respect to equilibrium, as shown in Fig. S1 b where we compare the ARPES spectrum with and without the presence of the pump.
The situation is slightly more complex at negative delays. The electron is photoemitted first by the probe pulse; when the pump pulse impinges on the sample after some delay ∆t, the outgoing photoelectron at the distance v∆t from the sample, where v is its velocity, is pushed by F P V . The smaller the delay, the stronger is the force felt, resulting in a gradual increase in the kinetic energy of the electrons at negative delays up to the time zero. These effects are clearly observed in time resolved spectra taken over long timescales (hundreds of picoseconds), see Fig. S1c,d. * mmichiardi@phas.ubc.ca † damascelli@physics.ubc.ca 2 In order to provide a consistent picture of the electronic dispersion in the material and to extract information of electron transport properties, the data in the main text are plotted against the non-equilibrium electron quasi-Fermi level (E F n ). To extract the E F n , we fit the energy distribution curve (EDC) integrated around the Fermi wave-vector of the topological surface state (red area in Fig. S1b) with a Fermi-Dirac distribution function. This is possible because the momentum-integrated spectral function of a linearly dispersing band corresponds to its density of states [4,5], and it allows us to reliably obtain the location of the non-equilibrium chemical potential of the sample at each time delay. Figure.

II. LIFETIME OF THE PHOTOVOLTAGE EFFECT ON 2DEGS
We discussed the effect of the PV on the energy and spin splitting of the quantum well states (QWSs) in the main text. The lifetime of the PV can be extracted from the time-resolved trace at BZ center in effect to be 950 ps ± 50 ps by fitting QWS1. Since QWS2 is above the chemical potential at negative delays, we impose the decay timescale extracted from QWS1, and find that QWS2 is ≈ 20 meV above the Fermi level at equilibrium. This is consistent with the observation that at negative delays there is no spectral signature of QWS2. We remark that the PV lifetime depends on experimental conditions -such as the spot size of the pump beam -as its value is governed by the diffusion dynamics of carriers from the illuminated area.

III. TEMPORAL DYNAMICS OF K F S AND ELECTRON DENSITY
The photovoltage effect introduces additional electrons on the surface, which causes the QWSs minima to drop to more negative values as detailed in the manuscript and shown in  Fig.S2. Both Rashba-split branches are shown for QWS1 in blue, while the splitting is too small for QWS2 to be analyzed within the experimental resolution. The uncertainty is largest between 0 and 2 ps, as the sharpness and intensity of features at E F is adversely affected by the direct optical excitation and increased electron temperature, respectively.
For this reason, extraction of k F of QWS2 in the 0 -2 ps time domain is not possible.
However, the k F of both QWSs clearly increase at large positive delays, and the effect is the largest for QWS2 -an expected consequence of the softening of the band bending potential (see Fig.S1 and S4). Using Luttinger's theorem and under the reasonable assumption that the Fermi surfaces are circular, we can use the following formula to extract the electron density per unit cell from the values of k F : Where A 2D BZ is the size of the 2-dimensional Brillouin Zone of Bi 2 Se 3 (111) and n e is the electron degeneracy of the state. The electron density increases by 2.4 ± 0.5 · 10 −3 (electrons/Unit cell) in QWS2 4 ps after the optical excitation. The value is smaller for QWS1 with 1.3 ± 0.5 · 10 −3 and it is negligible for the topological surface state.

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To simulate the band bending, we build a one dimensional model along the direction perpendicular to the surface; this is justified under the assumption that the lateral dimensions of interest (here 300 µm, the size of the probe beam) is much larger than the perpendicular one (≈ 50 nm, the width of the space charge region). We employ the modified Thomas-Fermi approximation (MTFA) approach, following the steps described in Ref. 6.
The calculation of the charge densities and electron potential normally requires to solve the coupled Poisson and Schrödinger equations self-consistently, a process that is numerically expensive. However, the MTFA allows to calculate both quantities from the Poisson equation through the introduction of a term, f MTFA , that accounts for the potential barrier at the surface, and reflects the surface confinement-induced quantization of the density of states.
To calculate the band energy as a function of depth x (where x = 0 corresponds to the surface), one needs to solve the Poisson equation in one dimension: for the electron potential V (x). The charge ρ in the semiconductor is a function of the concentrations of ionized donor and acceptor atoms in the bulk, N + D and N − A , as well as the electron and hole density distributions, n(x) and p(x). These latter values are calculated from the conduction and valence band density of states, g C/V , with the introduction of f MTFA (x) as: where F(E) is the Fermi-Dirac distribution, and E CB/VB is the energy of the bottom/top of the conduction/valence band. The term f MTFA (x) is given by: where E g is the band gap, k B is the Boltzmann constant, and T is the temperature.   The material specific parameters we used are summarized in Table I We set the surface boundary condition V 0 = −0.54 eV to simulate the results of the experiment described in Fig. 4 of the main text. This value was extracted by measuring the shift in energy of the TSS Dirac point between a freshly cleaved sample and the chemically gated sample. Thanks to the surface-nature of the TSS, its energy rigidly shifts with the surface potential. The energy profile of V (x) is displayed in Fig. S4a with a solid black line.
The envelope wave-functions of the QWSs are plotted at their respective energies, revealing the ladder-structure of the QWSs.
The electric field inside the material is obtained from V (x) as E = − dV (x) dx , and its depth profile is plotted in Fig. S4 b. The value of the Rashba spin-orbit coupling constant α, on the right axes, is calculated from the electric field using the equation where E g is the material band gap [13]. The spin-orbit energy splitting parameter ∆ is here used as a fitting parameter to reproduce the Rashba splitting of QWS1 in equilibrium. The 9 estimated value of 0.84 eV is later used for all calculations of the Rashba SOC in this system.

STATES IN PUMP-PROBE EXPERIMENTS
To calculate the time-dependent photovoltaic effect on the system under study, one needs to know the spatial distribution of the photo-generated charges, the electron-hole recombination rate, and the carriers diffusion coefficient [14]. In the effort to capture the main findings of our experiments, we work under the following approximations: (1) The diffusion rate is omitted in our calculations; the timescale for the carrier diffusion, extracted from our data, is 950 ps, and has little relevance within the short timescale (< 10 ps) of interest when reproducing the data shown in Fig. 4 in the main text. (2) The electronhole recombination is taken into account by the introduction of an effective photo-carrier density, N hν . After photo-carriers are generated, a large percentage recombines in a short time through electron-electron and electron-phonon scattering processes. Only a fraction of photo-generated carriers is swept apart by the electric field without recombining. This percentage of carriers constitute the effective photo-carrier density responsible for the PV effect.
(3) Finally, the photo-carrier spatial distribution is rationalized by employing a center of mass approach: the charge distribution is replaced with a single point charge Q n/p = ∓e·N hν situated at its center of mass, x n and x p for electron and holes respectively. The charge carriers' center of mass moves in time within the electric field akin a point charge as: where µ is the carrier mobility. Finally, analogously to a capacitor model, one can obtain the time-dependent equation for the PV magnitude [15]: In our simulation, we calculate the band bending from Eq. S2 at each time step using the boundary condition V 0 (t) = V 0 (0) + V PV (t). The QWS and Rashba splittings are also calculated at each step through Eqs. S6 and S7. The E F n at the surface is calculated under the flat quasi-Fermi level approximation [16] by finding the value of E F n that satisfies charge neutrality: with respect to E F n , one obtains the plot of Fig. 4c of the main text.

VII. CHEMICAL GATING DEPENDENCE OF THE PHOTOVOLTAGE EFFECT
To elucidate the relation between initial surface doping and the PV effect, we progressively evaporate K on a sample of Bi 2 Se 3 and measure the photoemission spectrum before the pump arrival (i.e., at a negative time delay of -1 ps), and 20 ps after the pump arrival. At 20 ps, the system is fully thermalized and bears the signature of the PV effect, namely an increase in binding energy due to the excess electron population at the surface. The change in binding energy, calculated as the difference between the QWS energy minima at ∆ = 20 and -1 ps, is presented in Fig. S6; it clearly decreases upon increasing the K deposition time, indicating a reduction of the PV effect for lager initial band bending potentials. To understand this result, we consider the ratio of the photo-charges responsible for the PV and the initial electron population in the QWSs. As chemical gating increases, the QWS's electron occupation and density of states increase, while the number of photo-charges giving rise to the PV effect stays constant. Thus, the photo-charges can be accommodated with a smaller binding energy shift, such that at the highest doping QWS1 shows only an infinitesimal energy shift.

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The Rashba spin-orbit coupling -a direct consequence of the change in the electric field in the SCR -will scale in the same way. Hence, the optical manipulation of 2DEGs will have a larger impact the smaller the surface gating.